No, you couldn't use a cup to the clean the bottom, that's why I suggested a turkey baster or an empty syringe for that part
If you could get a bit of airline tubing (cheap in any fish store) you can also start a syphon by placing one end in the fish jug, putting the other end lower down and sucking on it until water comes through. It's a skill that takes practice to get the syphon started without getting a mouthful of dirty tank water... chances are that you will, the first few times! Then you move the airline around the bottom, sucking up all the waste. This is drop the water level at the same time.
Yes, ammonia will build up in there, depends when the last water change was. I suggest a 50% change at first because a 100% change is drastic, and big sudden changes can again, put fish into shock and kill them. Small and often would be better when we don't know when he last had a water change.